Question

a cake has a circumference of ( 25\frac{1}{7} ) inches. what is the area of the top of the cake? use ( \frac{22}{7} ) for ( pi ). write your answer as a decimal and round to the nearest hundredth of a square inch.

Explanation:

Step1: Recall the formula for circumference of a circle

The formula for the circumference \( C \) of a circle is \( C = 2\pi r \), where \( r \) is the radius. We know \( C = 25\frac{1}{7}=\frac{25\times7 + 1}{7}=\frac{176}{7} \) inches and \( \pi=\frac{22}{7} \). We can solve for \( r \).

From \( C = 2\pi r \), we get \( r=\frac{C}{2\pi} \). Substituting the values:

\( r=\frac{\frac{176}{7}}{2\times\frac{22}{7}} \)

Simplify the denominator: \( 2\times\frac{22}{7}=\frac{44}{7} \)

So \( r = \frac{176}{7}\div\frac{44}{7}=\frac{176}{7}\times\frac{7}{44}=4 \) inches.

Step2: Recall the formula for area of a circle

The formula for the area \( A \) of a circle is \( A=\pi r^{2} \). We know \( \pi = \frac{22}{7} \) and \( r = 4 \) inches.

Substitute the values: \( A=\frac{22}{7}\times4^{2}=\frac{22}{7}\times16=\frac{352}{7}\approx50.29 \) square inches (rounded to the nearest hundredth).

Answer:

\( 50.29 \) square inches